Polarimetric observations of Fast Radio Bursts (FRBs) are a powerful resource for better understanding these mysterious sources by directly probing the emission mechanism of the source and the magneto-ionic properties of its environment. We present a pipeline for analysing the polarized signal of FRBs captured by the triggered baseband recording system operating on the FRB survey of The Canadian Hydrogen Intensity Mapping Experiment (CHIME/FRB). Using a combination of simulated and real FRB events, we summarize the main features of the pipeline and highlight the dominant systematics affecting the polarized signal. We compare parametric (QU-fitting) and non-parametric (rotation measure synthesis) methods for determining the Faraday rotation measure (RM) and find the latter method susceptible to systematic errors from known instrumental effects of CHIME/FRB observations. These errors include a leakage artefact that appears as polarized signal near $rm{RMsim 0 ; rad , m^{-2}}$ and an RM sign ambiguity introduced by path length differences in the systems electronics. We apply the pipeline to a bright burst previously reported by citet[FRB 20191219F;][]{Leung2021}, detecting an $mathrm{RM}$ of $rm{+6.074 pm 0.006 pm 0.050 ; rad , m^{-2}}$ with a significant linear polarized fraction ($gtrsim0.87$) and strong evidence for a non-negligible circularly polarized component. Finally, we introduce an RM search method that employs a phase-coherent de-rotation algorithm to correct for intra-channel depolarization in data that retain electric field phase information, and successfully apply it to an unpublished FRB, FRB 20200917A, measuring an $mathrm{RM}$ of $rm{-1294.47 pm 0.10 pm 0.05 ; rad , m^{-2}}$ (the second largest unambiguous RM detection from any FRB source observed to date).
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